Pseudorandom generators and learning algorithms forAC ⁰

Abstract

For anyAC ⁰ functionf ofn bits, there is a polynomialp such that anyp(logn)-wise decomposable distribution “fools”f. In other words,f cannot distinguish between the pseudorandom strings in the distribution and truly random strings. The polynomialp depends only on the size and depth of the circuit computingf.

This subsumes and extends the class of distributions that were previously known to foolAC ⁰ functions, and partially answers an open question posed by Linial and Nisan in 1990, as to whether every polylog-wise independent distribution foolsAC ⁰ functions or not.

Each polylog-wise decomposable distribution serves as a fixed training set of examples for learning (approximately interpolating) allAC ⁰ functions computed by circuits of some fixed depth and size. Furthermore, small, natural distributions (training sets) exist that yield deterministic learning algorithms that run in timeO (2^polylogn) forAC ⁰ functions, where the degree of the polylog depends on the size and depth of the circuit to be learnt.

This improves on the randomized algorithms with the same time complexity given, for example, by Linialet al. in 1989, where the examples for the training set are picked randomly from specific distributions.